I find in general that the pull-back of a section of a vector bundle is a section of the pull-back bundle, but this seems to be false for the cotangent bundle.
Let $\phi:M\to N$ be a smooth map, $\omega\in \Gamma(T^*N)$ be a $1$-form and $X\in \Gamma(TM)$ be a vector field.
Then, since $(\phi^*\omega)(X)=\omega(\phi_*X)$, we can say that $\phi^*\omega$ is a $1$-form of $T^*M\ncong\phi^*T^*N$, in particular the fibers of $T^*N$ and $T^*M$ are different and thus the pull-back bundle $\phi^*T^*N$ cannot be equivalent to $T^*M$.
$\textbf{My question}$: why?